Just a guess as I can't see your circuit, but this is one way sometimes used to suppress the inductive kickback when current to the coil is turned off. I'd have to see the circuit to give a more definite answer.
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No.
If it is a light emitting diode it is there to let you know that the relay is energized. In my experience, when a DC current is removed from a coil, the polarity of the magnetic field reverses and if it weren't for the diode, the relay would "chatter". The diode polarity is oppsite of the actuating voltage.
Quenching diodes are used to protect the control circuit of a relay. Power is applied to a relay's coil to close it. When this power is released there is a backwards flowing pulse that can wipe out solid state devices like transistors, LEDs, chips, etc... The quenching diode is applied in parallel with the relay coil so that the cathode is on the positive side. During normal operation, the diode doesn't conduct and the relay operates normally. When the relay is switched off, the "reversed" back pulse is absorbed due to the low voltage drop of the diode which limits the spike.
No, because as soon as a load is connected across a capacitor it will discharge. You need a continuous voltage supply across the coil of a relay to keep it energized. Yes. A large capacitor, 10000 mfd, can hold open a small relay for a few seconds, plus or minus, depending on your varibles.
under internal fault conditions , CT primary current produces steep insrease in voltage across the relay. This will exceed the burden 3kv normally designed as per. To get protected from the voltage, metrosil is connected across the relay , to drop the voltage.